Question: Captain Emily has a ship, the H.M.S. Khan. The ship is two furlongs from the dread pirate Nadia and her merciless band of thieves. The Captain has probability $\dfrac{1}{2}$ of hitting the pirate ship. The pirate only has one good eye, so she hits the Captain's ship with probability $\dfrac{1}{3}$. If both fire their cannons at the same time, what is the probability that both the pirate and the Captain hit each other's ships?
Solution: If the Captain hits the pirate ship, it won't affect whether she's also hit by the pirate's cannons (and vice-versa), because they both fired at the same time. So, these events are independent. Since they are independent, in order to get the probability that both the pirate and the Captain hit each other's ships, we just need to multiply together the probability that the captain hits and the probability that the pirate hits. The probability that the Captain hits is $\dfrac{1}{2}$ The probability that the pirate hits is $\dfrac{1}{3}$ So, the probability that both the pirate and the Captain hit each other's ships is $\dfrac{1}{2} \cdot \dfrac{1}{3} = \dfrac{1}{6}$.